Question

There are two identical cubes. Out of one cube, a sphere of maximum volume (V_S) is cut out. Out of the second cube, a cone of maximum volume (V_C) is cut such that its base lies on one of the faces of the cube. Which one of the following is correct?

• A. V_S = V_C
• B. V_S = 2V_C
• C. 2V_SV_S = 3V_C
• D. 3V_S = 4V_C

Answer 1

The correct option is B

V_S = 2V_C

Let the side of each cube be '$a$' units.

Then the radius of the biggest sphere that can be cut out
$=\frac{a}{2}units$

$\text{Volume of the sphere V}{}_S=\frac{4}{3}\times \pi \times (\frac{a}{2}{)}^3$
$=\frac{\pi a^3}{6}$

Similarly, radius of the base of the cone with maximum radius $=\frac{a}{2}units$,
height = $a$ units.

$\text{Maximum volume of the cone V}{}_C=\frac{1}{3}\times \pi \times (\frac{a}{2}{)}^2\times a$
$V_C=\frac{\pi a^3}{12}$

$\therefore V_S=2V_C$